For example, if φ=α∨βφαβ\varphi=\alpha\vee\beta, then αα\alpha and ββ\beta are subformulas of φφ\varphi. This is so because α∨β=¬⁡(¬⁢α∧¬⁢β)αβαβ\alpha\vee\beta=\neg(\neg\alpha\wedge\neg\beta), so that ¬⁢α∧¬⁢βαβ\neg\alpha\wedge\neg\beta is a subformula of φφ\varphi by applications of 1 followed by 2 above. By 3 above, ¬⁢αα\neg\alpha and ¬⁢ββ\neg\beta are subformulas of φφ\varphi. Therefore, by 2 again, αα\alpha and ββ\beta are subformulas of φφ\varphi.

For another example, if φ=∃x(∃y(x2+y2=1))fragmentsφxfragmentsnormal-(yfragmentsnormal-(superscriptx2superscripty21normal-)normal-)\varphi=\exists x(\exists y(x^{2}+y^{2}=1)), then ∃y(t2+y2=1)fragmentsyfragmentsnormal-(superscriptt2superscripty21normal-)\exists y(t^{2}+y^{2}=1) is a subformula of φφ\varphi as long as ttt is a term that does not contain the variable yyy. Therefore, if t=y+2ty2t=y+2, then ∃y((y+2)2+y2=1)fragmentsyfragmentsnormal-(superscriptfragmentsnormal-(y2normal-)2superscripty21normal-)\exists y((y+2)^{2}+y^{2}=1) is not a subformula of φφ\varphi. In fact, if y∈ℝyℝy\in\mathbb{R}, the equation(y+2)2+y2=1superscripty22superscripty21(y+2)^{2}+y^{2}=1 is never true.

Finally, it is easy to see (by induction) that if αα\alpha is a subformula of ψψ\psi and ψψ\psi is a subformula of φφ\varphi, then αα\alpha is a subformula of φφ\varphi. “Being a subformula of” is a reflexivetransitive relation on LLL-formulas.

Remark. There is also the notion of a literal subformula of a formula φφ\varphi. A formula ψψ\psi is a literal subformula of φφ\varphi if it is a subformula of φφ\varphi obtained in any one of the first three ways above, or if ∃x⁢(ψ)xψ\exists x(\psi) is a literal subformula of φφ\varphi.

Note that any literal subformula of φφ\varphi is a subformula of φφ\varphi, for if φ=∃x⁢(ψ)φxψ\varphi=\exists x(\psi), then xxxoccurs free in ψψ\psi and ψ=ψ⁢[x/x]ψψxx\psi=\psi[x/x].

In the second example above, ∃y(x2+y2=1)fragmentsyfragmentsnormal-(superscriptx2superscripty21normal-)\exists y(x^{2}+y^{2}=1) and x2+y2=1superscriptx2superscripty21x^{2}+y^{2}=1 are both literal subformulas of φ=∃x(∃y(x2+y2=1))fragmentsφxfragmentsnormal-(yfragmentsnormal-(superscriptx2superscripty21normal-)normal-)\varphi=\exists x(\exists y(x^{2}+y^{2}=1)).